Under Dirichlet boundary condition $u|_{x=0}=0$;
Under Neumann boundary condition $u_x|_{x=0}=0$;
Under Robin boundary condition $(u_x-ku)|_{x=0}=0$ ($k>0$)
$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\erf}{\operatorname{erf}}$ $\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$ $\newcommand{\arcsinh}{\operatorname{arcsinh}}$
For wave equation $$ u_{tt}-c^2u_{xx}=0\qquad \text{as } x>0 $$ we consider reflection of the wave from the boundary at $x=0$
Under Dirichlet boundary condition $u|_{x=0}=0$;
Under Neumann boundary condition $u_x|_{x=0}=0$;
Under Robin boundary condition $(u_x-ku)|_{x=0}=0$ ($k>0$)
In all these cases $$u(x,t)= \phi (t+c^{-1}x) +\psi(t-c^{-1}x)$$ with $\psi(t)=-\psi(t)$ for Dirichlet and $\psi(t)=\psi(t)$ for Neumann boundary conditions. For Robin boundary condition $\psi (t)$ is defined as a solution to ODE $\psi ' + ck\psi =\phi'-ck\phi$. We call $\phi (t+c^{-1}x)$ incoming wave and $\psi (t-c^{-1}x)$ outgoing (or reflected) wave.